Silver gelatin is the usual regular B&W printing paper.
Cibachrome is a colour process using colour dyes which work subtractively.
Baryta and fibre are one and the same type of paper. Resin coated paper is the newer type of paper which requires less washing time than fibre.

For B&W, another important distinction is graded vs. variable contrast (VC; aka multigrade or various other terms or contractions of them). Graded papers are designed to produce a single contrast grade and can't be adjusted, except to a limited extent by developer choice. VC papers, OTOH, include blue- and green-sensitive layers that respond differently, so you can adjust the amount of contrast in a print by varying the amount of blue vs. green light used when exposing the papers. This is typically done via magenta and yellow filters, either built into the enlarger or bought separately.

There are also differences in "tone." Cool-tone papers tend to have a faint blue or green tint, whereas warm-tone papers tend slightly toward the brown or red. Any paper's tone can be changed by toning, which is a separate (optional) processing step that's done after the basic processing.

If you're just starting out printing, VC RC papers are the easiest way to get started, but some people prefer fiber-based and/or graded papers. All of these are available in a variety of finishes (glossy, matte, etc.), sizes, tones, etc. If you've got negatives that you want to be professionally printed without doing the work yourself, ask the lab or shop to show you samples of the papers and processes they offer.